The function f(x)=1−1−x2 is

Continuous on [ − 1 , 1 ] and differentiable on ( − 1 , 0 ) ∪ ( 0 , 1 )

The function f(x)=1−1−x2 is

Continuous on [ − 1 , 1 ] and differentiable on ( − 1 , 0 ) ∪ ( 0 , 1 )

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$The function f (x) = \sqrt{1 - \sqrt{1 - x^{2}}} is$

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$Continuous as well as differentiable on [- 1, 1]$

$Continuous on [- 1, 1] and differentiable on (- 1, 0) \cup (0, 1)$

$Continuous on [- 1, 1] and differentiable on (- 1, 1)$

None of these

answer is B.

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Detailed Solution

$\begin{array}{l} f (x) = \sqrt{1 - \sqrt{1 - x^{2}}} \\ Clearly, the domain of definition is [- 1, 1] . \\ f' (x) = \frac{1}{2 \sqrt{1 - \sqrt{1 - x^{2}}}} (- \frac{1}{2 \sqrt{1 - x^{2}}}) (- 2 x) \\ = \frac{x}{\sqrt{1 - x^{2}} \sqrt{1 - \sqrt{1 - x^{2}}}}, \\ Which does not exist for x= 0, - 1, 1 \\ Also, f(x) is continuous on [- 1, 1] \\ So, (b) is the correct choice . \end{array}$